From many body quantum dynamics to nonlinear dispersive PDEs, and back

University Of Texas At Austin, Austin TX

Investigators

Abstract

The investigator plans to study existence and regularity of nonlinear dispersive PDEs as well as derivation of these equations. More precisely, the investigator proposes to study two groups of problems. The first group focuses on analyzing regularity of solutions to the super-critical nonlinear wave (NLW) and Schrodinger (NLS) equations. The last two decades brought numerous advances in understanding global existence of solutions to the so called "critical" nonlinear PDEs, where criticality is understood in the sense that a PDE possesses a quantity globally controlled in time which has the same regularity as a certain scaling invariant norm. However obtaining global in time solutions to super-critical equations remains a challenging problem. Here by a super-critical equation we mean that the conserved quantities are at lower regularities than the scaling invariant norm. A famous example is the 3D Navier-Stokes equations that describe the most fundamental properties of viscous incompressible fluids. Other examples involve various nonlinear wave equations that appear in the context of general relativity as well as Schrodinger equations. With her collaborators, the investigator proposes to work on three projects towards obtaining partial regularity results for super-critical NLW and NLS inspired by similar results available in the context of the 3D Navier-Stokes. The second group of problems focuses on projects related to derivation of the NLS from many body quantum dynamics. The investigator, together with Chen, proposes to develop further the work that they recently started on the Cauchy problem for the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDEs, that appear in the derivation of the NLS. The GP hierarchy describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE. Based on these dispersive features, the investigator proposes to investigate solutions to the GP hierarchy and illustrate that, in some instances, the GP can be studied using generalizations of methods of dispersive PDEs. Suggested problems involve important mathematical questions such as existence and regularity of solutions to PDEs that describe various wave phenomena. For instance, the NLS and their combinations with the Korteweg-de-Vries and wave equations have been proposed as models for many basic wave phenomena. Due to their physical significance, it is essential to develop tools to understand behavior of solutions to these nonlinear equations and the investigator plans to work in that direction via adapting some tools from her earlier work on equations of fluid motion (such as Navier-Stokes equations that describe fundamental properties of viscous fluids) to the context of dispersive equations. On the other hand, the investigator plans to continue her recent work on physically inspired questions related to derivation of dispersive PDEs from many body Boson systems. The proposed activity contains an interdisciplinary approach in the sense that it has potential to bring dispersive PDE methods to the level of many body quantum dynamics and vise versa. In particular, the long term goal is to try to adapt some of the recent advances from dispersive PDEs to the many body systems, where one has physically relevant questions that are beyond the reach of known mathematical methods.